01-平均差「Mean Deviation」

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  • 数据分析
  • 数据分析
大约 7 分钟

各个数据值离平均值平均有多远

怎样计算

先求数据的平均值……接着求距离……最后求距离的平均!

三部曲:

  • 一、求所有数据值的平均值open in new window
  • 二、求每个数据与平均值的距离(用数据值减去平均值,不理正负号)
  • 三、求这些距离的平均

像这样:

例子:3、6、6、7、8、11、15、16 的 平均差

一、求平均:

平均=3+6+6+7+8+11+15+168=728=9\huge\frac{3 + 6 + 6 + 7 + 8 + 11 + 15 + 16}{8}=\frac{72}{8}=9

二、求每个数据只与平均值的距离:

image-20221009232139634
与 9 的距离
36
63
63
72
81
112
156
167

像这样:

平均差

(没有正负号!)

三、求这些距离的平均

平均差=6+3+3+2+1+2+6+78=308=3.75\huge\frac{6 + 3 + 3 + 2 + 1 + 2 + 6 + 7}{8}=\frac{30}{8}=3.75

所以,平均值 = 9平均差 = 3.75

平均差告诉我们,每个数据值平均离中心有多远。

在上面的例子里,数据值平均离中心 3.75 远。

在这里,就是距离

公式

公式是:

平均差 = xμN\huge\frac{\sum|x−μ|}{N}

我们详细看看:

绝对偏差

在公式里求的每个距离是叫绝对偏差,因为它是偏差的绝对值open in new window(离平均值的距离)

绝对偏差

我们把 "|" 放在项的两变来代表 "绝对值":

|-3| = 3

为每一个 x

绝对偏差 = |x - μ|

在这个例子里,16绝对偏差 = |x - μ| = |16 - 9| = |7| = 7

现在把它们加起来……

总和符号

"加起来" 的符号是 Σ(叫总和符号open in new window),所以:

绝对偏差的总和 = Σ|x - μ|

除以数据的个数 N,得到:

平均差 =xμN\huge\frac{\sum|x−μ|}{N}

我们再用正确的符号做一次:

例子: 3、6、6、7、8、11、15、16 的 平均差

一、求平均:

μ=3+6+6+7+8+11+15+168=728=9\hugeμ=\frac{3+6+6+7+8+11+15+16}{8}=\frac{72}{8}=9

二、求绝对偏差:

image-20221010084338645

三、求平均差:

平均差 = xμN=308=3.75\huge\frac{\sum|x-μ|}{N}=\frac{30}{8}=3.75

注意:平均差有时也叫平均绝对偏差(英语 Mean Absolute Deviation (MAD)),因为它是绝对偏差的平均。

平均差的意思是什么?

平均差告诉我们,每个数据值平均离中心有多远。

以下是个例子(用 标准差open in new window 网页里的数据):

例子:你和你的朋友测量了你们的狗的身高(毫米):

狗统计图

身高(到肩膀)是:600mm、470mm、170mm、430mm 和 300mm。

一、求平均:

μ=600+470+170+430+3005=19705=394\hugeμ=\frac{600+470+170+430+300}{5}=\frac{1970}{5}=394

二、求绝对偏差:

image-20221010085924952
x|x - μ|
600206
47076
170224
43036
30094
Σ|x - μ| = 636

三、求平均差:

平均差=xμN=6365=127.2\huge\frac{\sum|x-μ|}{N}=\frac{636}{5}=127.2

所以,狗的身高与平均值的平均距离是 127.2 mm

(可以和 147 mm标准差open in new window比较一下)

检测

在平均值一边的偏差应该和另一边的偏差是相等的。

上面的例子:

例子:3、6、6、7、8、11、15、16

偏差是:

平均差

6 + 3 + 3 + 2 + 1 = 2 + 6 + 7

15 = 15

同样:

例子:狗

平均左边的偏差:224 + 94 = 318

平均右边的偏差:206 + 76 + 36 = 318

如果它们不相同……你便是做错了!

Question

Question 1

Calculate the mean deviation for the numbers: 75, 83, 96, 100, 121 and 125

A. 14.33

B. 15

C. 15.33

D. 16

Q1 Answer

Step 1 Find the mean:

μ = (75 + 83 + 96 + 100 + 121 + 125) ÷ 6 = 600 ÷ 6 = 100

Step 2 Find the Absolute Deviations:

[cached] 0.02598762512207 ms

Step 3 Find the Mean Deviation:

[cached] 0.0061988830566406 ms

Question 2

Ten friends scored the following marks in their end-of-year math exam: 23%, 37%, 45%, 49%, 56%, 63%, 63%, 70%, 72% and 82%

What was the mean deviation of their marks?

A. 14%

B. 15%

C. 16%

D. 17%

Q2 Answer

Step 1 Find the mean:

μ = (23 + 37 + 45 + 49 + 56 + 63 + 63 + 70 + 72 + 82) ÷ 10 = 560 ÷ 10 = 56%

Step 2 Find the Absolute Deviations:

[cached] 0.043869018554688 ms

Step 3 Find the Mean Deviation:

[cached] 0.030040740966797 ms

Question 3

A booklet has 12 pages with the following numbers of words: 271, 354, 296, 301, 333, 326, 285, 298, 327, 316, 287 and 314

What is the mean deviation of the number of words per page?

A. 18.5

B. 19.33

C. 20

D. 20.17

Q3 Answer

Step 1 Find the mean:

μ = (271 + 354 + 296 + 301 + 333 + 326 + 285 + 298 + 327 + 316 + 287 + 314) ÷ 12 = 3,708 ÷ 12 = 309

Step 2 Find the Absolute Deviations:

[cached] 0.045061111450195 ms

Step 3 Find the Mean Deviation:

[cached] 0.010013580322266 ms

Question 4

What is the mean deviation of the first 10 numbers of the Fibonacci sequence?

A. 8.8

B. 8.5

C. 8.42

D. 8.32

Q4 Answer

The first 10 numbers of the Fibonacci sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34

Step 1 Find the mean:

μ = (0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34) ÷ 10 = 88 ÷ 10 = 8.8

Step 2 Find the Absolute Deviations:

[cached] 0.060796737670898 ms

Step 3 Find the Mean Deviation:

[cached] 0.016927719116211 ms

Question 5

The Lakers scored the following numbers of goals in their last twenty matches: 3, 0, 1, 5, 4, 3, 2, 6, 4, 2, 3, 3, 0, 7, 1, 1, 2, 3, 4, 6

Calculate the Mean Deviation.

A. 1.5

B. 2

C. 2.5

D. 3

Q5 Answer

Method 1

Step 1 Find the mean:

μ = (3 + 0 + 1 + 5 + 4 + 3 + 2 + 6 + 4 + 2 + 3 + 3 + 0 + 7 + 1 + 1 + 2 + 3 + 4 + 6) ÷ 20 = 60 ÷ 20 = 3

Step 2 Find the Absolute Deviations:

[cached] 0.056028366088867 ms

Step 3 Find the Mean Deviation:

[cached] 0.011920928955078 ms

Method 2 This can be done much more neatly using tables, as follows:

First complete a frequency table:

[cached] 0.010967254638672 ms

Then calculate the mean and mean deviation from the following table:

[cached] 0.010967254638672 ms

Question 6

Ramiro did a survey of the number of pets owned by his classmates, with the following results:

[image]

What was the mean deviation of the number of pets?

A. 1.02

B. 1.053

C. 1.087

D. 1.12

Q6 Answer

Calculate the mean and mean deviation from the following table:

[cached] 0.063896179199219 ms

Question 7

Emma rolled a die a number of times and recorded her results in a bar graph, as follows: [image]

What was the mean deviation of the scores?

A. 1.375

B. 1.5

C. 1.625

D. 1.875

Q7 Answer

First complete a frequency table:

[cached] 0.063896179199219 ms

Then calculate the mean and mean deviation from the following table:

[cached] 0.016927719116211 ms

Question 8

Rachel rolled a die forty five times with the following results:

[image]

What was the mean deviation of her scores?

A. 1.564

B. 1.609

C. 1.64

D. 1.653

Q8 Answer

Calculate the mean and mean deviation from the following table:

[cached] 0.051975250244141 ms

Question 9

A spinner is in the shape of a regular heptagon marked with the numbers 1 to 7.

Theodore spun the spinner 50 times and recorded his results in a table, as follows:

[image]

Calculate the mean deviation of the scores.

A. 1.6088

B. 1.7288

C. 1.7688

D. 1.8088

Q9 Answer

Calculate the mean and mean deviation from the following table:

[cached] 0.063180923461914 ms

Question 10

The children in a class did a survey of the number of siblings (brothers and sisters) each of them had. The results are recorded in the following table:

[image]

Calculate the mean deviation of the number of siblings.

A. 1.38

B. 1.41

C. 1.45

D. 1.48

Q10 Answer

Calculate the mean and mean deviation from the following table:

[cached] 0.046014785766602 ms

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